Josephson effects in twisted cuprate bilayers

Twisted bilayers of high-Tc cuprate superconductors have been argued to form topological phases with spontaneously broken time reversal symmetry T for certain twist angles. A key outstanding challenge is to identify unambiguous signatures of these topological phases in experiments. With this goal in mind we theoretically investigate a suite of Josephson phenomena between twisted cuprate layers. At intermediate twist angles we find an unusual nonmonotonic temperature dependence of the critical current, which we attribute to the unconventional sign structure of the d-wave order parameter. The onset of the T-broken phase near the 45 degrees twist is marked by a crossover from the conventional 2 pi -periodic Josephson relation J(phi) similar or equal to Jc sin phi to a pi -periodic function as the single-pair tunneling becomes dominated by a second-order cotunneling process. Despite this fundamental change, the critical current remains a smooth function of the twist angle theta and temperature T implying that a measurement of Jc alone will not be a litmus test for the T-broken phase. We show that clear signatures of the T-broken phase appear when Jc is measured in the presence of an applied magnetic field or radiofrequency drive: The resulting Fraunhofer oscillations acquire additional nodes and fractional Shapiro steps become visible. We discuss these results in light of recent experiments on twisted bilayers of the high-T-c cuprate superconductor Bi2Sr2CaCu2O8+delta.